Bringing Eqs. (17) and (18–21) into the differential equations in Eqs. (7–12) of the perturbations, the perturbations in amplitude and phase are decoupled into two equation sets as Display Formula
$\u2202\delta \rho +F\u2202t=cn[\u2212\u2202\delta \rho +F\u2202z\u2212\delta \rho +P\u221212(l0\u22123\gamma \u02dc|F\xaf|2)\delta \rho +F\u2212i\beta 22\u22022\delta \rho +F\u2202t2],$(22)
Display Formula$\u2202\delta \rho \u2212F\u2202t=cn[\u2202\delta \rho \u2212F\u2202z\u2212\delta \rho \u2212P\u221212(l0\u22123\gamma \u02dc|F\xaf|2)\delta \rho \u2212F\u2212i\beta 22\u22022\delta \rho =F\u2202t2],$(23)
Display Formula$\u2202\delta \rho +P\u2202t=\u221212F\xaf(\delta \rho 0+\delta \rho 2)\u221212\rho \xaf0\delta \rho +F\u221212\rho \xaf2\delta \rho \u2212F\u2212\delta \rho +PT2,$(24)
Display Formula$\u2202\delta \rho \u2212P\u2202t=\u221212F\xaf(\delta \rho 0+\delta \rho 2)\u221212\rho \xaf0\delta \rho \u2212F\u221212\rho \xaf2\delta \rho +F\u2212\delta \rho \u2212PT2,$(25)
Display Formula$\u2202\delta \rho 0\u2202t=2F\xaf(\delta \rho +P+\delta \rho \u2212P)+2P\xaf(\delta \rho +F+\delta \rho \u2212F)\u2212\delta \rho 0T1,$(26)
Display Formula$\u2202\delta \rho 2\u2202t=F\xaf(\delta \rho +P+\delta \rho \u2212P)+P\xaf(\delta \rho +F+\delta \rho \u2212F)\u2212\delta \rho 2Tg$(27)
and Display Formula$\u2202\delta \theta +F\u2202t=cn[\u2212\u2202\delta \theta +F\u2202z\u2212P\xafF\xaf\delta \theta +P\u221212(l0\u22123\gamma \u02dc|F\xaf|2)\delta \theta +F],$(28)
Display Formula$\u2202\delta \theta \u2212F\u2202t=cn[\u2202\delta \theta \u2212F\u2202z\u2212P\xafF\xaf\delta \theta \u2212P\u221212(l0\u22123\gamma \u02dc|F\xaf|2)\delta \theta \u2212F],$(29)
Display Formula$\u2202\delta \theta +P\u2202t=\u221212F\xafP\xaf(\rho \xaf0\delta \theta +F+\rho \xaf2\delta \theta \u2212F+\rho \xaf2\delta \theta 2)\u2212\delta \theta +PT2,$(30)
Display Formula$\u2202\delta \theta \u2212P\u2202t=\u221212F\xafP\xaf(\rho \xaf0\delta \theta \u2212F+\rho \xaf2\delta \theta +F\u2212\rho \xaf2\delta \theta 2)\u2212\delta \theta \u2212PT2,$(31)
Display Formula$\u2202\delta \theta 2\u2202t=P\xafF\xaf\rho \xaf2(\delta \theta \u2212F+\delta \theta \u2212P\u2212\delta \theta +F\u2212\delta \theta +P)\u2212\delta \theta 2Tg.$(32)